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## G = D5×C4⋊Dic3order 480 = 25·3·5

### Direct product of D5 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×C4⋊Dic3
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — D5×C4⋊Dic3
 Lower central C15 — C30 — D5×C4⋊Dic3
 Upper central C1 — C22 — C2×C4

Generators and relations for D5×C4⋊Dic3
G = < a,b,c,d,e | a5=b2=c4=d6=1, e2=d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 796 in 184 conjugacy classes, 84 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×4], C30 [×3], C2×C4⋊C4, C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊Dic3, C4⋊Dic3 [×3], C22×Dic3 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C6×D5 [×6], C2×C30, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×C4⋊Dic3, D5×Dic3 [×4], D5×C12 [×4], C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D5×C4⋊C4, C30.Q8 [×2], C5×C4⋊Dic3, C605C4, C2×D5×Dic3 [×2], D5×C2×C12, D5×C4⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C2×C4⋊C4, C4×D5 [×2], C22×D5, C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, Q8×D5, C2×C4⋊Dic3, D5×Dic3 [×2], C2×S3×D5, D5×C4⋊C4, D5×Dic6, D5×D12, C2×D5×Dic3, D5×C4⋊Dic3

Smallest permutation representation of D5×C4⋊Dic3
On 240 points
Generators in S240
(1 51 47 15 55)(2 52 48 16 56)(3 53 43 17 57)(4 54 44 18 58)(5 49 45 13 59)(6 50 46 14 60)(7 39 219 232 36)(8 40 220 233 31)(9 41 221 234 32)(10 42 222 229 33)(11 37 217 230 34)(12 38 218 231 35)(19 29 63 71 95)(20 30 64 72 96)(21 25 65 67 91)(22 26 66 68 92)(23 27 61 69 93)(24 28 62 70 94)(73 119 83 109 105)(74 120 84 110 106)(75 115 79 111 107)(76 116 80 112 108)(77 117 81 113 103)(78 118 82 114 104)(85 130 153 121 102)(86 131 154 122 97)(87 132 155 123 98)(88 127 156 124 99)(89 128 151 125 100)(90 129 152 126 101)(133 177 150 143 167)(134 178 145 144 168)(135 179 146 139 163)(136 180 147 140 164)(137 175 148 141 165)(138 176 149 142 166)(157 170 191 212 184)(158 171 192 213 185)(159 172 187 214 186)(160 173 188 215 181)(161 174 189 216 182)(162 169 190 211 183)(193 237 210 203 227)(194 238 205 204 228)(195 239 206 199 223)(196 240 207 200 224)(197 235 208 201 225)(198 236 209 202 226)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 227)(8 228)(9 223)(10 224)(11 225)(12 226)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(25 60)(26 55)(27 56)(28 57)(29 58)(30 59)(31 194)(32 195)(33 196)(34 197)(35 198)(36 193)(37 201)(38 202)(39 203)(40 204)(41 199)(42 200)(43 70)(44 71)(45 72)(46 67)(47 68)(48 69)(49 96)(50 91)(51 92)(52 93)(53 94)(54 95)(73 129)(74 130)(75 131)(76 132)(77 127)(78 128)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 120)(86 115)(87 116)(88 117)(89 118)(90 119)(103 156)(104 151)(105 152)(106 153)(107 154)(108 155)(109 126)(110 121)(111 122)(112 123)(113 124)(114 125)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 182)(140 183)(141 184)(142 185)(143 186)(144 181)(145 160)(146 161)(147 162)(148 157)(149 158)(150 159)(163 216)(164 211)(165 212)(166 213)(167 214)(168 215)(169 180)(170 175)(171 176)(172 177)(173 178)(174 179)(205 220)(206 221)(207 222)(208 217)(209 218)(210 219)(229 240)(230 235)(231 236)(232 237)(233 238)(234 239)
(1 89 29 79)(2 90 30 80)(3 85 25 81)(4 86 26 82)(5 87 27 83)(6 88 28 84)(7 164 224 214)(8 165 225 215)(9 166 226 216)(10 167 227 211)(11 168 228 212)(12 163 223 213)(13 123 93 73)(14 124 94 74)(15 125 95 75)(16 126 96 76)(17 121 91 77)(18 122 92 78)(19 115 55 100)(20 116 56 101)(21 117 57 102)(22 118 58 97)(23 119 59 98)(24 120 60 99)(31 141 201 188)(32 142 202 189)(33 143 203 190)(34 144 204 191)(35 139 199 192)(36 140 200 187)(37 134 194 184)(38 135 195 185)(39 136 196 186)(40 137 197 181)(41 138 198 182)(42 133 193 183)(43 153 67 103)(44 154 68 104)(45 155 69 105)(46 156 70 106)(47 151 71 107)(48 152 72 108)(49 132 61 109)(50 127 62 110)(51 128 63 111)(52 129 64 112)(53 130 65 113)(54 131 66 114)(145 205 170 230)(146 206 171 231)(147 207 172 232)(148 208 173 233)(149 209 174 234)(150 210 169 229)(157 217 178 238)(158 218 179 239)(159 219 180 240)(160 220 175 235)(161 221 176 236)(162 222 177 237)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)(109 110 111 112 113 114)(115 116 117 118 119 120)(121 122 123 124 125 126)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)(145 146 147 148 149 150)(151 152 153 154 155 156)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)(181 182 183 184 185 186)(187 188 189 190 191 192)(193 194 195 196 197 198)(199 200 201 202 203 204)(205 206 207 208 209 210)(211 212 213 214 215 216)(217 218 219 220 221 222)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)
(1 149 4 146)(2 148 5 145)(3 147 6 150)(7 156 10 153)(8 155 11 152)(9 154 12 151)(13 134 16 137)(14 133 17 136)(15 138 18 135)(19 161 22 158)(20 160 23 157)(21 159 24 162)(25 172 28 169)(26 171 29 174)(27 170 30 173)(31 132 34 129)(32 131 35 128)(33 130 36 127)(37 126 40 123)(38 125 41 122)(39 124 42 121)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 144 52 141)(50 143 53 140)(51 142 54 139)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 214 70 211)(68 213 71 216)(69 212 72 215)(73 194 76 197)(74 193 77 196)(75 198 78 195)(79 209 82 206)(80 208 83 205)(81 207 84 210)(85 232 88 229)(86 231 89 234)(87 230 90 233)(91 186 94 183)(92 185 95 182)(93 184 96 181)(97 218 100 221)(98 217 101 220)(99 222 102 219)(103 224 106 227)(104 223 107 226)(105 228 108 225)(109 204 112 201)(110 203 113 200)(111 202 114 199)(115 236 118 239)(116 235 119 238)(117 240 120 237)

G:=sub<Sym(240)| (1,51,47,15,55)(2,52,48,16,56)(3,53,43,17,57)(4,54,44,18,58)(5,49,45,13,59)(6,50,46,14,60)(7,39,219,232,36)(8,40,220,233,31)(9,41,221,234,32)(10,42,222,229,33)(11,37,217,230,34)(12,38,218,231,35)(19,29,63,71,95)(20,30,64,72,96)(21,25,65,67,91)(22,26,66,68,92)(23,27,61,69,93)(24,28,62,70,94)(73,119,83,109,105)(74,120,84,110,106)(75,115,79,111,107)(76,116,80,112,108)(77,117,81,113,103)(78,118,82,114,104)(85,130,153,121,102)(86,131,154,122,97)(87,132,155,123,98)(88,127,156,124,99)(89,128,151,125,100)(90,129,152,126,101)(133,177,150,143,167)(134,178,145,144,168)(135,179,146,139,163)(136,180,147,140,164)(137,175,148,141,165)(138,176,149,142,166)(157,170,191,212,184)(158,171,192,213,185)(159,172,187,214,186)(160,173,188,215,181)(161,174,189,216,182)(162,169,190,211,183)(193,237,210,203,227)(194,238,205,204,228)(195,239,206,199,223)(196,240,207,200,224)(197,235,208,201,225)(198,236,209,202,226), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,227)(8,228)(9,223)(10,224)(11,225)(12,226)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(25,60)(26,55)(27,56)(28,57)(29,58)(30,59)(31,194)(32,195)(33,196)(34,197)(35,198)(36,193)(37,201)(38,202)(39,203)(40,204)(41,199)(42,200)(43,70)(44,71)(45,72)(46,67)(47,68)(48,69)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95)(73,129)(74,130)(75,131)(76,132)(77,127)(78,128)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,120)(86,115)(87,116)(88,117)(89,118)(90,119)(103,156)(104,151)(105,152)(106,153)(107,154)(108,155)(109,126)(110,121)(111,122)(112,123)(113,124)(114,125)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,182)(140,183)(141,184)(142,185)(143,186)(144,181)(145,160)(146,161)(147,162)(148,157)(149,158)(150,159)(163,216)(164,211)(165,212)(166,213)(167,214)(168,215)(169,180)(170,175)(171,176)(172,177)(173,178)(174,179)(205,220)(206,221)(207,222)(208,217)(209,218)(210,219)(229,240)(230,235)(231,236)(232,237)(233,238)(234,239), (1,89,29,79)(2,90,30,80)(3,85,25,81)(4,86,26,82)(5,87,27,83)(6,88,28,84)(7,164,224,214)(8,165,225,215)(9,166,226,216)(10,167,227,211)(11,168,228,212)(12,163,223,213)(13,123,93,73)(14,124,94,74)(15,125,95,75)(16,126,96,76)(17,121,91,77)(18,122,92,78)(19,115,55,100)(20,116,56,101)(21,117,57,102)(22,118,58,97)(23,119,59,98)(24,120,60,99)(31,141,201,188)(32,142,202,189)(33,143,203,190)(34,144,204,191)(35,139,199,192)(36,140,200,187)(37,134,194,184)(38,135,195,185)(39,136,196,186)(40,137,197,181)(41,138,198,182)(42,133,193,183)(43,153,67,103)(44,154,68,104)(45,155,69,105)(46,156,70,106)(47,151,71,107)(48,152,72,108)(49,132,61,109)(50,127,62,110)(51,128,63,111)(52,129,64,112)(53,130,65,113)(54,131,66,114)(145,205,170,230)(146,206,171,231)(147,207,172,232)(148,208,173,233)(149,209,174,234)(150,210,169,229)(157,217,178,238)(158,218,179,239)(159,219,180,240)(160,220,175,235)(161,221,176,236)(162,222,177,237), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,149,4,146)(2,148,5,145)(3,147,6,150)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,134,16,137)(14,133,17,136)(15,138,18,135)(19,161,22,158)(20,160,23,157)(21,159,24,162)(25,172,28,169)(26,171,29,174)(27,170,30,173)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,144,52,141)(50,143,53,140)(51,142,54,139)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,214,70,211)(68,213,71,216)(69,212,72,215)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,209,82,206)(80,208,83,205)(81,207,84,210)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,186,94,183)(92,185,95,182)(93,184,96,181)(97,218,100,221)(98,217,101,220)(99,222,102,219)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,204,112,201)(110,203,113,200)(111,202,114,199)(115,236,118,239)(116,235,119,238)(117,240,120,237)>;

G:=Group( (1,51,47,15,55)(2,52,48,16,56)(3,53,43,17,57)(4,54,44,18,58)(5,49,45,13,59)(6,50,46,14,60)(7,39,219,232,36)(8,40,220,233,31)(9,41,221,234,32)(10,42,222,229,33)(11,37,217,230,34)(12,38,218,231,35)(19,29,63,71,95)(20,30,64,72,96)(21,25,65,67,91)(22,26,66,68,92)(23,27,61,69,93)(24,28,62,70,94)(73,119,83,109,105)(74,120,84,110,106)(75,115,79,111,107)(76,116,80,112,108)(77,117,81,113,103)(78,118,82,114,104)(85,130,153,121,102)(86,131,154,122,97)(87,132,155,123,98)(88,127,156,124,99)(89,128,151,125,100)(90,129,152,126,101)(133,177,150,143,167)(134,178,145,144,168)(135,179,146,139,163)(136,180,147,140,164)(137,175,148,141,165)(138,176,149,142,166)(157,170,191,212,184)(158,171,192,213,185)(159,172,187,214,186)(160,173,188,215,181)(161,174,189,216,182)(162,169,190,211,183)(193,237,210,203,227)(194,238,205,204,228)(195,239,206,199,223)(196,240,207,200,224)(197,235,208,201,225)(198,236,209,202,226), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,227)(8,228)(9,223)(10,224)(11,225)(12,226)(13,64)(14,65)(15,66)(16,61)(17,62)(18,63)(25,60)(26,55)(27,56)(28,57)(29,58)(30,59)(31,194)(32,195)(33,196)(34,197)(35,198)(36,193)(37,201)(38,202)(39,203)(40,204)(41,199)(42,200)(43,70)(44,71)(45,72)(46,67)(47,68)(48,69)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95)(73,129)(74,130)(75,131)(76,132)(77,127)(78,128)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,120)(86,115)(87,116)(88,117)(89,118)(90,119)(103,156)(104,151)(105,152)(106,153)(107,154)(108,155)(109,126)(110,121)(111,122)(112,123)(113,124)(114,125)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,182)(140,183)(141,184)(142,185)(143,186)(144,181)(145,160)(146,161)(147,162)(148,157)(149,158)(150,159)(163,216)(164,211)(165,212)(166,213)(167,214)(168,215)(169,180)(170,175)(171,176)(172,177)(173,178)(174,179)(205,220)(206,221)(207,222)(208,217)(209,218)(210,219)(229,240)(230,235)(231,236)(232,237)(233,238)(234,239), (1,89,29,79)(2,90,30,80)(3,85,25,81)(4,86,26,82)(5,87,27,83)(6,88,28,84)(7,164,224,214)(8,165,225,215)(9,166,226,216)(10,167,227,211)(11,168,228,212)(12,163,223,213)(13,123,93,73)(14,124,94,74)(15,125,95,75)(16,126,96,76)(17,121,91,77)(18,122,92,78)(19,115,55,100)(20,116,56,101)(21,117,57,102)(22,118,58,97)(23,119,59,98)(24,120,60,99)(31,141,201,188)(32,142,202,189)(33,143,203,190)(34,144,204,191)(35,139,199,192)(36,140,200,187)(37,134,194,184)(38,135,195,185)(39,136,196,186)(40,137,197,181)(41,138,198,182)(42,133,193,183)(43,153,67,103)(44,154,68,104)(45,155,69,105)(46,156,70,106)(47,151,71,107)(48,152,72,108)(49,132,61,109)(50,127,62,110)(51,128,63,111)(52,129,64,112)(53,130,65,113)(54,131,66,114)(145,205,170,230)(146,206,171,231)(147,207,172,232)(148,208,173,233)(149,209,174,234)(150,210,169,229)(157,217,178,238)(158,218,179,239)(159,219,180,240)(160,220,175,235)(161,221,176,236)(162,222,177,237), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,149,4,146)(2,148,5,145)(3,147,6,150)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,134,16,137)(14,133,17,136)(15,138,18,135)(19,161,22,158)(20,160,23,157)(21,159,24,162)(25,172,28,169)(26,171,29,174)(27,170,30,173)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,144,52,141)(50,143,53,140)(51,142,54,139)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,214,70,211)(68,213,71,216)(69,212,72,215)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,209,82,206)(80,208,83,205)(81,207,84,210)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,186,94,183)(92,185,95,182)(93,184,96,181)(97,218,100,221)(98,217,101,220)(99,222,102,219)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,204,112,201)(110,203,113,200)(111,202,114,199)(115,236,118,239)(116,235,119,238)(117,240,120,237) );

G=PermutationGroup([(1,51,47,15,55),(2,52,48,16,56),(3,53,43,17,57),(4,54,44,18,58),(5,49,45,13,59),(6,50,46,14,60),(7,39,219,232,36),(8,40,220,233,31),(9,41,221,234,32),(10,42,222,229,33),(11,37,217,230,34),(12,38,218,231,35),(19,29,63,71,95),(20,30,64,72,96),(21,25,65,67,91),(22,26,66,68,92),(23,27,61,69,93),(24,28,62,70,94),(73,119,83,109,105),(74,120,84,110,106),(75,115,79,111,107),(76,116,80,112,108),(77,117,81,113,103),(78,118,82,114,104),(85,130,153,121,102),(86,131,154,122,97),(87,132,155,123,98),(88,127,156,124,99),(89,128,151,125,100),(90,129,152,126,101),(133,177,150,143,167),(134,178,145,144,168),(135,179,146,139,163),(136,180,147,140,164),(137,175,148,141,165),(138,176,149,142,166),(157,170,191,212,184),(158,171,192,213,185),(159,172,187,214,186),(160,173,188,215,181),(161,174,189,216,182),(162,169,190,211,183),(193,237,210,203,227),(194,238,205,204,228),(195,239,206,199,223),(196,240,207,200,224),(197,235,208,201,225),(198,236,209,202,226)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,227),(8,228),(9,223),(10,224),(11,225),(12,226),(13,64),(14,65),(15,66),(16,61),(17,62),(18,63),(25,60),(26,55),(27,56),(28,57),(29,58),(30,59),(31,194),(32,195),(33,196),(34,197),(35,198),(36,193),(37,201),(38,202),(39,203),(40,204),(41,199),(42,200),(43,70),(44,71),(45,72),(46,67),(47,68),(48,69),(49,96),(50,91),(51,92),(52,93),(53,94),(54,95),(73,129),(74,130),(75,131),(76,132),(77,127),(78,128),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,120),(86,115),(87,116),(88,117),(89,118),(90,119),(103,156),(104,151),(105,152),(106,153),(107,154),(108,155),(109,126),(110,121),(111,122),(112,123),(113,124),(114,125),(133,187),(134,188),(135,189),(136,190),(137,191),(138,192),(139,182),(140,183),(141,184),(142,185),(143,186),(144,181),(145,160),(146,161),(147,162),(148,157),(149,158),(150,159),(163,216),(164,211),(165,212),(166,213),(167,214),(168,215),(169,180),(170,175),(171,176),(172,177),(173,178),(174,179),(205,220),(206,221),(207,222),(208,217),(209,218),(210,219),(229,240),(230,235),(231,236),(232,237),(233,238),(234,239)], [(1,89,29,79),(2,90,30,80),(3,85,25,81),(4,86,26,82),(5,87,27,83),(6,88,28,84),(7,164,224,214),(8,165,225,215),(9,166,226,216),(10,167,227,211),(11,168,228,212),(12,163,223,213),(13,123,93,73),(14,124,94,74),(15,125,95,75),(16,126,96,76),(17,121,91,77),(18,122,92,78),(19,115,55,100),(20,116,56,101),(21,117,57,102),(22,118,58,97),(23,119,59,98),(24,120,60,99),(31,141,201,188),(32,142,202,189),(33,143,203,190),(34,144,204,191),(35,139,199,192),(36,140,200,187),(37,134,194,184),(38,135,195,185),(39,136,196,186),(40,137,197,181),(41,138,198,182),(42,133,193,183),(43,153,67,103),(44,154,68,104),(45,155,69,105),(46,156,70,106),(47,151,71,107),(48,152,72,108),(49,132,61,109),(50,127,62,110),(51,128,63,111),(52,129,64,112),(53,130,65,113),(54,131,66,114),(145,205,170,230),(146,206,171,231),(147,207,172,232),(148,208,173,233),(149,209,174,234),(150,210,169,229),(157,217,178,238),(158,218,179,239),(159,219,180,240),(160,220,175,235),(161,221,176,236),(162,222,177,237)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108),(109,110,111,112,113,114),(115,116,117,118,119,120),(121,122,123,124,125,126),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144),(145,146,147,148,149,150),(151,152,153,154,155,156),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180),(181,182,183,184,185,186),(187,188,189,190,191,192),(193,194,195,196,197,198),(199,200,201,202,203,204),(205,206,207,208,209,210),(211,212,213,214,215,216),(217,218,219,220,221,222),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240)], [(1,149,4,146),(2,148,5,145),(3,147,6,150),(7,156,10,153),(8,155,11,152),(9,154,12,151),(13,134,16,137),(14,133,17,136),(15,138,18,135),(19,161,22,158),(20,160,23,157),(21,159,24,162),(25,172,28,169),(26,171,29,174),(27,170,30,173),(31,132,34,129),(32,131,35,128),(33,130,36,127),(37,126,40,123),(38,125,41,122),(39,124,42,121),(43,164,46,167),(44,163,47,166),(45,168,48,165),(49,144,52,141),(50,143,53,140),(51,142,54,139),(55,176,58,179),(56,175,59,178),(57,180,60,177),(61,191,64,188),(62,190,65,187),(63,189,66,192),(67,214,70,211),(68,213,71,216),(69,212,72,215),(73,194,76,197),(74,193,77,196),(75,198,78,195),(79,209,82,206),(80,208,83,205),(81,207,84,210),(85,232,88,229),(86,231,89,234),(87,230,90,233),(91,186,94,183),(92,185,95,182),(93,184,96,181),(97,218,100,221),(98,217,101,220),(99,222,102,219),(103,224,106,227),(104,223,107,226),(105,228,108,225),(109,204,112,201),(110,203,113,200),(111,202,114,199),(115,236,118,239),(116,235,119,238),(117,240,120,237)])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 5 5 2 2 2 6 6 6 6 10 10 30 30 30 30 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + - + + + + + - + + + - - + - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 D5 Dic3 D6 D6 D6 D10 D10 Dic6 D12 C4×D5 S3×D5 D4×D5 Q8×D5 D5×Dic3 C2×S3×D5 D5×Dic6 D5×D12 kernel D5×C4⋊Dic3 C30.Q8 C5×C4⋊Dic3 C60⋊5C4 C2×D5×Dic3 D5×C2×C12 D5×C12 C2×C4×D5 C6×D5 C6×D5 C4⋊Dic3 C4×D5 C2×Dic5 C2×C20 C22×D5 C2×Dic3 C2×C12 D10 D10 C12 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 2 2 2 4 1 1 1 4 2 4 4 8 2 2 2 4 2 4 4

Matrix representation of D5×C4⋊Dic3 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 43 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 18 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 59 0 0 0 0 1 1
,
 60 60 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 17 53 0 0 0 0 36 44 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 13 17 0 0 0 0 26 48

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,18,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,59,1],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,36,0,0,0,0,53,44,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,13,26,0,0,0,0,17,48] >;

D5×C4⋊Dic3 in GAP, Magma, Sage, TeX

D_5\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("D5xC4:Dic3");
// GroupNames label

G:=SmallGroup(480,488);
// by ID

G=gap.SmallGroup(480,488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^6=1,e^2=d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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